SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response.
1) List the assumptions for testing hypotheses that three or more means are equivalent.

1) The populations have approximately normal distributions.
2) The populations have the same variance \({\sigma ^2}\) (or standard deviation \(\sigma \)).
3) The samples are random and independent of each other.
4) The different samples are from populations that are categorized in only one way. (The requirements of normality and equal variances are somewhat relaxed.)

2) The test statistic for one-way ANOVA is F = (variance between samples)/(variance within samples). Describe variance within samples and variance between samples. What relationship between variance within samples and variance between samples would result in the conclusion that the value of F is significant?

Variance between samples measures the variation between the sample means of the groups treated differently, that is the variation due to the treatment. The variance within the samples depends solely on the sample variances of the groups treated alike. The F ratio compares the two. If the F ratio is relatively close to 1, the two variances are about the same, and we conclude that there are no significant differences among the sample means. When the value of F is excessively large (that is, greater than 1), we conclude that the variation among the samples is not the same and that the means are not equal.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Given below are the analysis of variance results from a Minitab display. Assume that you want to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean.

3)

Source

DF

SS

MS

F

p

Factor

3

30

10.00

1.6

0.264

Error

8

50

6.25

Total

11

80

Find the critical value.
○ 7.59
○ 8.85
○ 1.6
● 4.07

4)

Source

DF

SS

MS

F

p

Factor

3

30

10.00

1.6

0.264

Error

8

50

6.25

Total

11

80

Identify the p-value.
○ 1.6
○ 10.00
○ 6.25
● 0.264

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance.
5) The data below represent the weight losses for people on three different exercise programs.

Exercise A

Exercise B

Exercise C

2.5

5.8

4.3

8.8

4.9

6.2

7.3

1.1

5.8

9.8

7.8

8.1

5.1

1.2

7.9

At the 1% significance level, does it appear that a difference exists in the true mean weight loss produced by the three exercise programs?

Test statistic: F = 1.491. Critical value: F = 6.927.
Fail to reject the claim of equal means. The data do not provide sufficient evidence to conclude that there is a difference in the true mean weight loss produced by the three exercise programs.

6) At the 0.025 significance level, test the claim that the three brands have the same mean if the
following sample results have been obtained.

Brand A

Brand B

Brand C

32

27

22

34

24

25

37

33

32

33

30

22

36

21

39

H0: \({\mu _1}\) = \({\mu _2}\) = \({\mu _3}\). H1: The means are not all equal.
Test statistic: F = 12.1230. Critical value: F = 5.0959.
Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the three brands have the same mean.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
7) Fill in the missing entries in the following partially completed one-way
ANOVA table.

Source

df

SS

MS=SS/df

F

Treatment

21.1

Error

20

3.1

Total

25

●

Source

df

SS

MS=SS/df

F

Treatment

5

21.1

4.22

1.36

Error

20

62.0

3.1

Total

25

83.1

○

Source

df

SS

MS=SS/df

F

Treatment

45

21.1

0.47

306.29

Error

20

62.0

3.1

Total

25

83.1

○

Source

df

SS

MS=SS/df

F

Treatment

5

21.1

4.22

0.73

Error

20

62.0

3.1

Total

25

21.26

○

Source

df

SS

MS=SS/df

F

Treatment

5

21.1

4.22

1.36

Error

20

62.0

3.1

Total

25

83.1

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

8) Use the data given below to verify that the t test for independent samples and the ANOVA
method are equivalent.

A

B

85

74

81

72

73

65

91

83

64

59

i) Use a t test with a 0.05 significance level to test the claim that the two samples come from populations with the same means.
ii) Use the ANOVA method with a 0.05 significance level to test the same claim.
iii) Verify that the squares of the t test statistic and the critical value are equal to the F test statistic and critical value.